Given: |x + 3| + |x - 2| = 4|x+3|+|x−2|=4
Subtract |x+3||x+3| from both sides:
|x - 2| = 4-|x + 3|" [1]"|x−2|=4−|x+3| [1]
Take a minute to look at application of the definition:
|A| = {(A; A >=0),(-A; A < 0):}
|x - 2| = {(x - 2; x -2 >=0),(2-x; x -2 < 0):}
|x + 3| = {(x + 3; x +3 >=0),(-x-3; x +3 < 0):}
Simplifying the inequalities:
|x - 2| = {(x - 2; x >= 2),(2-x; x < 2):}
|x + 3| = {(x + 3; x >=-3),(-x-3; x < -3):}
Substitute the case x >=2 into equation [1]:
x - 2 = 4-(x + 3);x >=2
x - 2 = 4-x - 3;x >=2
2x = 4 - 3+ 2;x >=2
2x = 3;x >=2
x = 3/2; x >=2 larr This solution is outside the domain, therefore, it is invalid.
Substitute the case -3<=x<2 into equation [1]:
2-x = 4-(x + 3);-3<=x<2
2-x = 4-x - 3;-3<=x<2
2 = 1;-3<=x<2 larr no solution
Substitute the case x < -3 into equation [1]:
2-x = 4-(-x-3);x < -3
2-x = 4+x+3;x < -3
-2x = 4+3-2;x < -3
-2x = 5;x < -3
x = -5/2;x < -3 larr This solution is outside the domain, therefore, it is invalid.
The overall answer is that there is no solution.
I checked this with WolframAlpha and it agreed with me.